I am reading the research article "The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets" by Shishikura. The following two definitions are given without any examples in this paper. Therefore to understand the following definitions, I need examples of these two definitions:
Let $f$ be rational map on $\mathbb{C}_{\infty}$. A closed subset $X$ of $\mathbb{C}_{\infty}$ is called a hyperbolic subset for $f$ if
- $f(X) \subset X$ and
- there exist a positive constant $c$ and $\kappa$ $> 1$ such that $\lVert (f^n)^{'} \rVert \geq c \kappa^{n}$ on $X$ for $n \geq 0$. Here $\rVert . \rVert$ denotes the norm of derivative with respect to the spherical metric on $\mathbb{C}_{\infty}$.
Let $\Lambda$ be open set of $\mathbb{C}$. A family $\{f_\lambda : \lambda \in \Lambda \}$ of rational maps is $J$-Stable at $\lambda_0 \in \Lambda$, if there exists a continuous map $h : \Lambda ^{'} \times J(f_{\lambda_{0}}) \rightarrow \mathbb{C}_{\infty}$, such that $\Lambda ^{'}$ is neighborhood of $\lambda_0$ in $\Lambda$, $h_{\lambda} \equiv h(\lambda,.)$ is conjugacy from $(J(f_{\lambda_{0}}),f_{\lambda_{0}})$ to $(J(f_{\lambda}), f_{\lambda})$ and $h_{\lambda_{0}} = \mathrm{id}_{J(f_{\lambda_{0}})}$.
Where $J(f)$ denotes the Julia set of $f$ and in both definitions $\mathbb{C}_{\infty}$ denotes the Riemann sphere.
To get the example of second definition I tried to work with the quadratic family $P(z) = z^2 + c$. But didn't get anything.
Note that I am not aware about hyperbolic dynamics. After studying some general theory of complex dynamics and some great examples in it. I am studying this research article to understand the proof that Hausdorff dimension of the boundary of the Mandelbrot set is $2$.
